Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

TL;DR

This paper explores relationships among Fourier coefficients of degree 2 Siegel modular forms with arbitrary level, extending previous work by establishing new relations under certain eigenform conditions for Hecke operators.

Contribution

It generalizes existing results to forms of arbitrary level and character, providing new relations among Fourier coefficients when the form is an eigenform for specific Hecke operators.

Findings

Established relations among Fourier coefficients for degree 2 Siegel forms.

Extended previous results to forms with arbitrary level and character.

Provided conditions under which these relations hold for eigenforms.

Abstract

We extend some recent work of D. McCarthy, proving relations among some Fourier coefficients of a degree 2 Siegel modular form $F$ with arbitrary level and character, provided there are some primes $q$ so that $F$ is an eigenform for the Hecke operators $T(q)$ and $T_1(q^2)$.